copula functions

There are two limitations in the analysis of drought characteristics, which this study has investigated and resolved. First, the limitation of the length of the statistical period regarding the presentation of meteorological drought characteristics and the other is the frequency analysis. The first case was solved by using CRU climate data and the second case by using copula functions in this research. In this study, while checking the accuracy of the rainfall values extracted from the CRU climate model on a monthly scale, the frequency analysis of drought severity and duration characteristics based on the SPI index in the Zarinehrood basin has been analyzed. The results of the investigations showed that the values of the CRU scenario have a suitable accuracy and error rate with the observational data and have a suitable certainty. By extending the statistical period to 60 years, the SPI index was estimated in the studied area, which indicates the increase in the severity and duration of droughts in recent years. 45% of the studied statistical period had a lack of rainfall and 8% of the studied months were faced with severe drought. By choosing the distribution of generalized extreme values and logistic for drought severity and duration series in the studied stations, Frank's copula was selected for drought severity-duration pair variable. The results of investigation and joint analysis of drought severity -duration pair variables led to the presentation of drought probability curves in the region, which estimate the regional characteristics of drought with different probabilities. The presented approach has better performance due to the increase of time series memory and the use of joint distribution and copula functions and shows the characteristics of drought better.

From a hydrological point of view, measuring the flow of rivers, lakes and groundwater is a measure of drought and there is a baseline time between the lack of rainfall and the decrease of running water of inlets and lakes and groundwater. More studies have been done on meteorological droughts compared to hydrological droughts. Therefore, the purpose of this study is multivariate analysis of hydrological droughts in Lake Urmia basin using artificial data generation models and Copula functions. Therefore, using a combination of the above methods for the analysis of hydrological droughts was used as a new method for the analysis of hydrological droughts.

In this study, in order to multivariate analysis of hydrological droughts in the Urmia Lake basin, the flow data of 28 hydrometric stations in which the flow regime is real were used during a statistical period of 40 years (1978-2017). Also, Ar (1) model was used to generate artificial data and SDImod index was used for drought analysis. For this purpose, artificial data were generated in 1000 sequence. Since univariate drought analysis and analysis based on historical data can not show the horizontal of future droughts alone, so using the Ar (1) model, annual data were generated and then using the model The Valencia and Schakke generated monthly artificial data. Then drought characteristics (intensity and duration) were extracted for both historical and generation data series and common distributions in hydrology were fitted to intensity, duration and flow data. Then the transfer probability matrix and their steady state condition matrix (SSC) were also calculated. Also, multivariate analysis of hydrological droughts was performed using ten Archimedean Copula functions. The above coding was done in MATLAB software environment.

The results of this study showed that after examining the homogeneity of data and their static test, most of the data had the necessary homogeneity and the results of data homogeneity showed that the coefficient of explanation was above 0.9 and the results of static test and Their trend showed that the data were within the allowable range of 1.2 ±2.1 and ±1.96. The results of fitting the data on the common statistical distributions showed that the Log Pearson Type3 (LP3) function was known as the superior distribution functions on the flow data and the gamma and exponential distribution functions on the severity and duration of the drought, respectively. The number of drought periods based on different scales of SDImod index showed that for different periods the number of drought periods for short-term scales was more than long-term scales. Also, the average intensity and duration of drought for generated and historical data indicate an increase in the intensity of drought for generated data compared to historical data. The results of classifying drought periods for historical and generated data showed that approximately 68% of the data were in the normal range during the statistical period and 32% were other classes. The result of the Copula functions showed that the Joe Copula function in the first order and Filip Gumble and Galambos functions in the next order were known as the superior Copula functions.

Finally, the results showed that the artificial data generation models for annual and monthly data for statistical years less than 30 years maintain the statistical characteristics of mean, standard deviation, skewness and correlation between two consecutive months, while increasing The number of statistical years of model performance becomes more favorable. The cumulative probability of non-annual drought and the probability of normal and wet season in hot months of the year is higher than other months of the year. Also, with increasing periods of drought, the cumulative probability of non-drought increases, so that with increasing periods, this probability decreases and becomes almost zero. The results of the joint and conditional return periods as well as the Kendall return period showed that the probability of drought occurring in future periods is expected to be at least similar to the historical data. The results also showed that the Joe Copula function was recognized as the superior Copula function for historical and generated data. Accordingly, the theoretical Copula function is close to the 45 degree angle bisector against the experimental Copula function.

Climate Climate change is one of the important factors that will affect different parts of human life on the planet and will have detrimental effects on the environment, socio-economic, and especially water resources. Knowledge of climate change can provide comprehensive plans in various areas of management regarding the monitoring of droughts and their potential risks. Drought can occur in any area, even wetlands. This phenomenon depends on various factors and parameters and one of the most important symbols of this phenomenon is the occurrence of drought is a decrease in rainfall and therefore the analysis of precipitation data is of special importance to study drought. The purpose of this study is to analyze drought variables using SPI and SPImod indices and detailed functions.

 In this study, to model the multivariate analysis of drought in Lake Urmia basin using RCP8.5 and RCP4.5 representative concentration pathway scenarios, data and models of atmospheric circulation of historical data (1991-2010) for three near horizons (2030- 2011), medium (2065-2046) and round (2099-2080) were simulated and produced. Then, using SPImod index and copula functions, drought multivariate analysis was performed in MATLAB software environment. In general, first, using the mentioned indicators (two indicators, SPI and SPImod), the characteristics of drought intensity and duration were extracted, then, using coding in MATLAB software environment, eight families of Archimedean detailed functions were used.

The results of multivariate analysis showed that the Joe copula function is the best copula function for drought multivariate analysis (For analysis of both severity and duration of drought for the study area). Also, the results of probability and the joint return period showed that in the coming periods, at least droughts of the same level as historical droughts and even more severe will occur. Thus, by studying the period of combined and conditional returns and Kendall, the results showed that at a certain critical probability level, the amount of Kendall return period is much more than the standard return period, so that this difference increases with increasing that certain amount.

The results obtained with the climate change approach on the meteorological drought of Lake Urmia showed that in the coming periods we will see an increase in temperature, which will affect the rate of trade in the region and water resources, on the other hand, because the data Meteorology and hydrology are used to calculate the types of droughts, so droughts affected by climate change will be so that in future periods 46% to 48% of the months will be dry in different horizons. Finally, the results of the time series of indicators showed that during the statistical period at least 40% of the months were dry and this intensity of droughts in the Urmia station is much higher than others. The modified SPI largely eliminates the disadvantages of conventional SPIs and takes into account seasonal variations in precipitation in the calculation of the SPI index.

Precipitation information plays an important role in calculating aquifer nutrition using mathematical models. In recent years, with the availability of satellite precipitation data, especially TRMM and GPM satellites, new and innovative methods have been developed to overcome the lack of access to precipitation data. However, barriers such as data uncertainty also limit these methods. In this study, after correcting the satellite data error, this information was used as a feed parameter to MODFLOW code, and the groundwater level uncertainty was calculated by different Coppola functions. Examination of groundwater model outputs showed a 50% reduction in root mean square error index (RMSE). It is worth noting that about 90% of the aquifer had a difference of less than 10%, about 8% had a difference of 20 to 30% and about 2% of the aquifer had an approximate difference of 80% of the observational data ratio. The mentioned results show the proper performance and with a reliability coefficient of over 90% of the Coppola functions in calculating the groundwater level uncertainty using satellite precipitation data as the feeding parameter.

Drought is a climatic phenomenon that slowly and gradually emerges, and is of a latent nature. It lasts long, damaging the different sectors of agriculture, environment and consequently the society. Monitoring and prediction of droughts, especially accurate determination of their times of emergence, and duration, are very important in water resources management and in planning for drought mitigation strategies. Throughout the present study, drought conditions in three provinces (Golestan, Guilan and Mazandaran) located in the southern margin of the Caspian Sea, were evaluated by means of Joint Deficit Index (JDI). The performance of JDI was compared with two other drought indices, Standardized Precipitation Index (SPI) and Modified Standardized Precipitation Index (SPImod). To follow the purpose, monthly precipitation data from 5 synoptic stations, namely: Babolsar, Bandareanzali, Ramsar, Rasht and Gorgan during the period of 1971 to 2011 were used for calculating the drought indices. Results showed that in recent years, the number of dry months across the study area had increased, significantly, as for all the considered stations (except Babolsar) the percentage of dry months had increased to more than 50% during the recent 10 years of 2002-2011. Based upon the calculated JDI, SPI and SPImod values, it becomes evident that the dry condition (along with deficit in precipitation) increase with an increase in the distance from the Caspian Sea. The results also indicate that JDI provides for a comprehensive assessment of droughts and that it is capable of reflecting both emerging and prolonging of the droughts in an accurate manner, allowing for a month-by-month drought assessment.

In the conventional methods of flood frequency analysis, the flood peak variable is just considered and assumed that this variable follows some specific parametric distribution function. This assumption would restrict us and lead us to the limited available information to evaluate the flood risk. It is well known that a flood event has three variables of flood peak, volume and duration which are random in nature and are mutually dependent. In this research, the concept of the Copula function is briefly introduced and then used to modeling the dependency structure of the flood variables of the Karun River at the Ahvaz hydrometric stationand then estimate their joint probability distribution. We use three well-known and appropriate copulas, including Ali–Mikhail–Haq, Cook–Johnson and Gumbel–Hougaard which belong to the Archimedean class of copulas, to modeling the joint probability distribution of the flood variables, where the marginal distributions of the flood’s variables are selected from the parametric and non-parametric distributions. The Gumbel–Hougaard family led to better modeling of different combination of flood’s variables based on goodness of fit criteria. The selected copula is used to estimate conditional cumulative distribution function and joint return periods which lead to better estimation of flood risk.

Univariate frequency analysis of hydrological events has some shortcomings caused by the lack of taking into account all characteristics of such events. Therefore, bivariate frequency analysis of hydrologic events such as flood can be useful in hydraulic design of structures and water resources management. The peak discharge and volume of flood are two important parameters in design of structures, thus, Archimedean copula functions were used for analysis of dependence structure between peak discharges and flood volumes. The time series of discharge for Arazkusehhydrometric station for a period of 40 years was constructed based on annual maximum (AM) discharge in daily scale. This station area is 1678.1 km2.The results showed that Gumbel extreme value copula was the best choice for fitting to data. Beased on design requirements, one can choose joint return period in "and", "or" and "conditional". For example, considering univariate frequency analysis results the return period of peak discharge equal to 50 years, while for a same value of peak discharge and volume in "and", "or" condition, the joint return periods are 72 and 38 respectively. The return period of "or" case was less than univariateand the highest value was belong to "and" case. This shows that planning or design based on "or" case is more confident, because results to larger values of peak discharge and volume quantile for the same value or univariate return period.